Questions tagged [ct.category-theory]
Enriched categories, topoi, abelian categories, monoidal categories, homological algebra.
1,781
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Birkhoff's HSP theorem in categories other than $\mathbf{Set}$
Fix a category $C$ with finite products and a set $L$ of function symbols (each equipped with an arity in $\mathbb N$).
An $L$-algebra in $C$, $\mathbf A=(A,(f^\mathbf{A})_{f\in L})$, is given by some ...
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268
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Which topoi are local with respect to Stone-Cech compactification?
Compact Hausdorff spaces $X$ are characterized among all topological spaces by the fact that for any topological space $S$, the embedding $S \to \beta S$ into its Stone-Cech compactification induces a ...
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368
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Describing the points of a glued topos
Let $f : \mathbf{X}\to \mathbf{Y}$ be a morphism of topoi; in his 1977 monograph, Johnstone describes the open mapping cylinder of $f$ as the following pushout of topoi:
$\require{AMScd}$
\begin{CD}
\...
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Categorification of "Every domain embeds into a field"?
In the category of commutative rings, every domain embeds into a field. Is this true in the category of presentably symmetric monoidal stable $\infty$-categories? Here's what I mean by that.
Let $...
13
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395
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The Barr-Boole-Galois topos; a modification of sets to play well with schemes
William Lawvere, in his 2015 CT talk "Alexander Grothendieck & the Concept of Space", introduced the "Barr-Boole-Galois topos", which plays the role that sets do for topological spaces (with ...
13
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433
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Examples of non-proper model structure
I have recently been thinking about left and right semi-model categories and in which case they can be promoted to Quillen model structure, and I have come to the conclusion that that absolutely all ...
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291
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Is $\mathrm{Hom}(P^i,P^j)$ a finite set? ($P=$ power set functor, $i\equiv j\bmod2$)
Let $P:\textbf{Set}\to\textbf{Set}$ be the contravariant power set functor, and put $P^n:=P\circ\cdots\circ P$ ($n$ factors), so that $P^n$ is a covariant (resp. contravariant) endofunctor of $\textbf{...
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Is every colimit-generator dense in an $\infty$-topos?
Recall that there are various senses in which a full subcategory $G \subseteq C$ may "generate" a category $C$. For example, in order of increasing strength (under reasonable conditions):
$G$ is a ...
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592
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Algebraic closure of a field in constructive mathematics
There is a short note by André Joyal called Les théorèmes de Chevalley-Tarski et remarque sur l'algèbre constructive (pp. 256-258). It is claimed there that there is a constructive version of the ...
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494
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How much choice is required to prove concretizability theorems in category theory?
A concretization of a category is a faithful functor to the category of sets. A category is concretizable if there exists such a functor.
An evident necessary condition for concretizability is ...
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910
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Categorification of finite type invariants
Recall that finite type invariants are those numerical knots invariants vanishing on knots with enough singularities. The value of an invariant on a singular knots is computed using the Vassiliev ...
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331
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Is there a common framework for Tannaka and Gabriel-Ulmer reconstruction theorems?
Gabriel-Ulmer duality is a biequivalence between the 2-category of finite limit categories and the 2-category of locally finitely presentable categories. It allows for the reconstruction of a theory ...
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178
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Large V-categories admitting the construction of V-presheaves
By a result of Foltz, and Freyd and Street, a category $C$ is essentially small (i.e. equivalent to a small category) if and only if both $C$ and $[C^{\text{op}}, \mathrm{Set}]$ are locally small. I ...
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Intuitionistic proofs of propositional formulae versus natural transformations between finite sets
The setup: Given a formula $\varphi$ of intuitionistic propositional logic (i.e., made from the connectors $\Rightarrow$, $\land$, $\lor$, $\top$ and $\bot$ from propositional variables $A,B,C,\ldots$)...
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Limits in free cocompletion, constructively
Classically, if a locally small category $C$ has all limits of shape $K$ (for some small diagram $K$), then its free co-completion also has $K$-shapped limits.
But all proof I know of that result ...
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Sentences preserved under inverse limits
One of the classical theorems of model theory is the Chang-Łos-Suszko preservation theorem that states that the theories formulated in FOL (first order logic) that are preserved under direct limits (...
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Differential rig categories
Note: this is a crossposting. The same thread appeared on the Category Theory channel in zulip, but since the latter requires access I can't really forward the message to people that do not have a ...
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Is there a bestiary of "derived 2-vector spaces"?
The appendix "A bestiary of 2–vector spaces" of Bartlett, Douglas, Schommer-Pries, Vicary, "Modular categories as representations of the 3-dimensional bordism 2-category" analyzes ...
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736
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Why do some tricks in homological algebra work over the category of C*-algebras?
The category of $C^*$-algebras is not abelian (a "proof" that it is pre-abelian can be found here, but it does not seem correct; I can't find any authoritative sources). However, it's ...
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Is the morphism of sheaves $(R \mapsto GL(R((h)))) \rightarrow (R \mapsto PGL(R((h))))$ surjective in Zariski topology?
Consider two functors given by $R \mapsto GL(R((h)))$ and $R \mapsto PGL(R((h)))$ for a ring $R$. It is easy to see that these functors are sheaves in Zariski topology (in fact for any affine variety $...
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380
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Looking for an invariant similar to algebraic K-theory
I'm wondering if there is an invariant, similar to algebraic K-theory, topological hochshild homologic, topological cyclic homology etc... that has the following properties:
a) It attach to each small ...
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378
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Which abelian categories have homological dimension 1?
In this MSRI lecture Geometry of Quiver Varieties I, Victor Ginzburg describes all abelian categories of homological dimension $1$ as being either
a category of representations $\mathrm{Rep}_\mathbf{...
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402
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The $\infty$-category of $n$-manifolds and open embeddings determined homotopically from that of topological manifolds?
Let $\mathrm{Diff}_n$, $\mathrm{PL}_n$, $\mathrm{Top}_n$ denote the $\infty$-categories of $n$-manifolds which are respectively smooth/PL/topological, and open embeddings (for instance by taking the ...
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381
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Is every Grothendieck category with a generator a category of sheaves?
The Gabriel-Popescu theorem tells us that every Grothendieck category with a generator is a left exact localization of a module category. I'm interested in a slightly different way of "representing" ...
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549
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What's wrong with the obvious argument that the unstable motivic category is an $\infty$-topos?
Fix a base scheme $S$, and let $Sm_S$ be the (ordinary) category of smooth schemes over $S$.
Denote by $Psh(Sm_S)$ the $\infty$-category of $\infty$-presheaves on $Sm_S$.
Let $Sh_{Nis}(Sm_S)\...
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410
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Biased vs unbiased lax monoidal categories
There are two principal ways to define a monoidal category:
The biased definition includes a unit object $I$, a binary tensor product $A\otimes B$, and a ternary associativity isomorphism $(A\otimes ...
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291
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Connections in terms of tangent ($\infty$-)categories?
Given a commutative ring $k$ and a commutative $k$-algebra $A$, we know that the Kähler differential $\Omega_{A/k}^1$ could be described through machineries of tangent categories (see, for example, ...
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422
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Abelian categories have become the language of homological algebra. Why haven't Zariski categories become the language of commutative algebra?
I'm not seeing much mention of Zariski categories in the literature. There is no article on Zariski categories in nLab, which would seem to be an obvious place to have such an article. What has ...
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689
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What is an homotopy group in a model category?
What is the notion, if any, of which all the known homotopy groups are particular cases?
Let me elaborate on this.
Given a model category $\cal M$ one can define a notion of homotopy group with $A$ ...
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274
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How can you unitalize a higher category?
Given an associative nonunital algebra $A$, there are (at least) two standard ways to produce a unital algebra $A'$ together with a map $A \to A'$. Following the discussion in the comments below, ...
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368
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Examples of Loci?
A (1-)locus is defined by Joyal to be a locally presentable category $C$ (generated by a small set of compact objects under colimits) with a zero object such that collections of objects in $C$ indexed ...
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330
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Does the $(\mathbb Z/2)$-graded isomorphism $E_n \cong E_{n+2}$ have any nice properties?
This question assumes everything is dg. Let's decide to work over the "field" $\mathbb Q[\mu,\mu^{-1}]$ where $\mu$ has homological degree $+2$. Then chain complexes are just $\mathbb Z/2$-graded. ...
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276
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Is there a Rado category?
The Rado graph appears to have a nice universality property (it contains all finite and all countably infinite graphs as induced subgraphs) and homogeinety property (any isomorphism between finite/...
12
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428
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Higher holonomies for higher local systems
In Jacob Lurie's classification of tqfts, one finds a version of the cobordism hypothesis for $(X,\zeta)$-structure, where an $(X,\zeta)$ structure on a manifold $M$ is the datum of a continuous map $...
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450
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What is the history of the notion of subdivision of categories?
A recent answer by Peter May prompts me to ask a question which I have been considering to ask for several months. (The reason why I have not asked it before is that it is not directly related to my ...
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The algebras and coalgebras of the homology functor
My question is very simple, but I suspect far from the intuition with which singular homology is introduced.
Consider singular homology as a functor
$$H_n : {\sf Top}\times{\sf Ab} \to \sf Ab$$
This ...
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104
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Description of the canonical equivalence between Adelman's free abelian category and Freyd's free abelian category on an additive category?
Let $\mathcal A$ be an additive category. Then there is a free abelian category $F(\mathcal A)$ on $\mathcal A$. I'm aware of two constructions in the literature, and I'd like to relate them.
The ...
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334
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A right adjoint preserves Phi-colimits if and only if the left adjoint does what?
Let $\Phi$ be a class of categories (e.g. filtered categories), and consider an adjunction $L : \mathbf C \rightleftarrows \mathbf D : R$. A $\Phi$-colimit is a colimit whose diagram is in $\Phi$. We ...
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356
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Category theory book with lots of examples
A student of mine is writing her Bachelor's thesis in which she has to use several categorical concepts (co/equalizers, co/products, subobjects, (regular) monomorphisms, etc.) in a couple of different ...
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137
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Stack completions in realizability toposes
An internal category $\mathbb A$ in an elementary topos $\mathcal E$ is called an (intrinsic) stack if the indexed category that it represents, $(X\in \mathcal E) \mapsto \mathcal{E}(X,\mathbb A) \in \...
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339
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Categorial foundations via "categories of algebras"
There are categorical foundations for mathematics axiomatizing the category of sets (Lawvere's ETCS), cartesian closed categories (type theory), and the category of spaces (homotopy type theory). ...
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296
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What is the core of a localization?
Let $(\mathbf{C}, \mathbf{W})$ be a relative category where $\mathbf{W}$ is saturated; i.e. it is equal to the subcategory of arrows inverted by $\gamma : \mathbf{C} \to \mathbf{C}[\mathbf{W}^{-1}]$.
...
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649
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Kontsevich's derived noncommutative geometry and Rosenberg's noncommutative 'spaces'
It appears to me (though I may be wrong) that the common opinion is that the main difference between derived noncommutative geometry and Rosenberg's noncommutative 'spaces' is that Rosenberg's version ...
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264
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Direct proof of the equivalence of symmetric monoidal $K$-theory and exact sequence $K$-theory?
When all exact sequences split in $C$, we have $\Omega B C \simeq K(C):=\Omega Q(C)$. Heuristically, this is because the space of upper-triangular matrices is contractible. Can this be made precise? I ...
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308
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Snake lemma for equivalence relation
A sequence $$E(\zeta) \stackrel{\theta}{\to} X^2 \rightrightarrows X \stackrel{\zeta}{\to} Y $$
where the unlabelled arrows are the two projection, is said to be exact iff
$\zeta$ is the ...
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197
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Every category is a localization of a concrete one?
$\require{AMScd}$Kučera, JPAA 1971 shows the remarkable result that every category is a quotient of a concrete one:
Given a category $\cal K$ there is a category $\check{\mathcal{K}}$ which is ...
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743
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How to compute Ext-groups for categories without enough injectives/projectives?
I am studying a category of representations of an infinite-dimensional Lie algebra, which is an abelian category. Unfortunately, the category does not have enough injectives/projectives. I would ...
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545
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The intrinsic meaning of abelian sheaf cohomology of a category
Basically my question is:
Suppose I meet an alien mathematician which understands everything through category theory and category theory alone. How would I convince said mathematician that certain ...
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477
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End of the Ext functors
Let $R$ be a ring, and consider the hom functor $\hom\colon Mod(R)^\text{op}\times Mod(R)\to Mod(R)$; the end of $\hom$ is well-known to be the set of endomorphisms (endonatural transformations) of ...
11
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290
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Goodwillie calculus and morphisms of functors
Let $F,G: \mathcal{T}\to \mathcal{S}$ be two functors from topological spaces to spectra (or topological spaces) and let $s: F\to G$ be a morphism between them.
Suppose $F$ and $G$ are analytic and ...